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International Journal of Bifurcation and Chaos, Vol. 28, No. 12 (2018) 1850144 (7 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218127418501444

Two Simplest Quadratic Chaotic MapsWithout Equilibrium

Shirin PanahiBiomedical Engineering Department,Amirkabir University of Technology,

Tehran 15875-4413, Iran

Julien C. SprottDepartment of Physics, University of Wisconsin,

Madison, WI 53706, USA

Sajad Jafari∗Biomedical Engineering Department,Amirkabir University of Technology,

Tehran 15875-4413, Iransajadjafari83@gmail.com

Received February 2, 2018; Revised May 11, 2018

Two simple chaotic maps without equilibria are proposed in this paper. All nonlinearities arequadratic and the functions of the right-hand side of the equations are continuous. The procedureof their design is explained and their dynamical properties such as return map, bifurcationdiagram, Lyapunov exponents, and basin of attraction are investigated. These maps belong tothe hidden attractor category which is a newly introduced category of dynamical system.

Keywords : Chaos; map; hidden attractors.

1. Introduction

Most known examples of chaotic flows have one ormore saddle points. Such saddle points allow homo-clinic and heteroclinic orbits and the prospect ofrigorously proving the chaos when the Shilnikovcondition is satisfied. One common way of locatingattractors of a dynamical system is to choose theinitial condition around its saddle points [Leonovet al., 2011; Kuznetsov et al., 2017]. Such attractorshave been called “self-excited,” and they are themost common type of dynamical systems describedin the literature [Leonov et al., 2012; Leonov &Kuznetsov, 2013; Leonov et al., 2014].

Recently, new chaotic flows have been discov-ered that are not associated with a saddle point.

These include the ones without any equilibriumpoints [Wei, 2011; Jafari et al., 2013], with onlystable equilibria [Wang & Chen, 2012; Molaie et al.,2013], or with a line containing infinitely many equi-librium points [Gotthans & Petržela, 2015; Jafari &Sprott, 2013]. The attractors for such systems iscalled “hidden attractors” [Leonov et al., 2015;Sharma et al., 2015; Danca & Kuznetsov, 2017],and that is because it is hard to discover them andthere is no systematic way to find initial conditionsthat lead to these attractors except by extensivenumerical search. There are plenty of researcheswhich are associated with designing and studyingrare examples of simple chaotic flows with hiddenattractors [Jafari et al., 2018; Pham et al., 2018;

∗Author for correspondence

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(a) (b)

Fig. 1. (a) Logistic map (for A = 4) and (b) a chaotic discontinuous map proposed in [Jafari et al., 2016a].

Tang et al., 2018]. However, there is little knowl-edge about this topic in chaotic maps and discretesystems [Jiang et al., 2016; Jafari et al., 2016a].

In this paper, we consider two-dimensional (2D)chaotic maps with no equilibria which are catego-rized as a system with hidden attractors [Jafariet al., 2016a]. The authors in [Jafari et al., 2016a]have introduced some chaotic maps with hiddenattractors for the first time. A different class oftwo-dimensional and three-dimensional maps withdifferent kinds of equilibria has been introduced in[Jiang et al., 2016].

It should be noted that, it is impossible tohave a one-dimensional “continuous” chaotic mapwith no equilibria. The reason is that the domainand range of these maps should cover each other;therefore there would be at least one point wherethe return map in the state space meets the iden-tity functions and this means there exists at leastone equilibrium. One example is the known Logisticmap [represented in Fig. 1(a)]. It should be notedthat Jafari et al. [2016a] proposed a 1D chaotic mapwith no equilibria. However that map was “discon-tinuous” [Fig. 1(b)].

In this paper, we introduce two simple chaotic2D No Equilibria Maps (NEM). It is easier to con-struct NEM with discontinuity in their equations,but they are of less interest [Sprott, 2010]. Thuswe focus on quadratic maps with no discontinu-ity in the right-hand equations. The rest of thepaper is organized as follows: Section 2 is aboutdesigning new simple 2D maps without equilibria.

Some conventional investigations of chaotic mapssuch as plotting strange attractors, bifurcation dia-gram, and Lyapunov exponents diagram are donein Sec. 3. Finally, Sec. 4 is the conclusion.

2. New No Equilibrium Maps

In this part, we perform a systematic search to find2D chaotic maps with no equilibria. We use ourown custom software which is described in [Sprott,2010]. The main point is to find the simplest noequilibrium chaotic maps with quadratic nonlinear-ities. We consider the following difference equationsas a class of 2D maps.

x(n + 1) = y(n) + x(n),

y(n + 1) = F (x(n), y(n)) + y(n)[a1x(n)

+ a2y(n) + a3] + y(n),

(1)

where a1, a2, and a3 are real coefficients and F (·)is a continuous function that determines the typeof system we want to design. In order to find thefixed points of System (1), the following conditionsshould be satisfied:

x(n) = y(n) + x(n) → y(n) = 0, (2)y(n) = F (x(n), y(n)) + y(n)[a1x(n)

+ a2y(n) + a3] + y(n)

→ F (x(n), y(n)) = 0. (3)Thus, to design the chaotic map with no equilib-rium, we should just choose F (x(n), y(n)) in a way

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Two Simplest Quadratic Chaotic Maps Without Equilibrium

Table 1. Simplest quadratic chaotic map with no equilibria.

Case Map Initial Condition Parameter

NEM1 x(n + 1) = y(n) + x(n)

y(n + 1) = 0.1x(n)2 + 0.1 + y(n)[−x(n) − by(n)] + y(n) (1.7,−0.39) b = 2NEM2 x(n + 1) = y(n)

y(n + 1) = −0.9y(n)2 + cx(n)y(n) + 1 (1, 0.1) c = 2

that it cannot become zero. If we choose the func-tion as second degree equation with ∆ < 0, thismap cannot have any real equilibria:

F (x(n), y(n)) = a4x(n)2 + a5x(n) + a6,

Condition : a25 − 4a4a6 < 0.(4)

Another simple structure which can be used indesigning 2D maps without equilibria, is Hénon-likemap. Thus we consider the following quadratic 2Dmap:

x(n + 1) = y(n),

y(n + 1) = a1x(n) + a2y(n) + a3x(n)2

+ a4y(n)2 + a5x(n)y(n) + a6.

(5)

To find the equilibria of this map the following con-ditions are needed:

x(n) = y(n),

y(n) = a1x(n) + a2y(n) + a3x(n)2

+ a4y(n)2 + a5x(n)y(n) + a6.

(6)

Thus, in order to find the equilibria, the followingequation should be solved:

(a3 + a4 + a5)x2 + (a1 + a2 − 1)x + a6 = 0. (7)To design a system without equilibria, it is enoughto find the coefficients of Eq. (7) which satisfy ∆ <0. As a result, the condition to find the map withoutequilibria is Eq. (8):

x(n) = y(n),

y(n) = a1x(n) + a2y(n) + a3x(n)2

+ a4y(n)2 + a5x(n)y(n) + a6,

Condition : (a1 + a2 − 1)2

− 4a6(a3 + a4 + a5) < 0.

(8)

We performed a systematic search to find chaoticsolutions in Systems (1) and (5) considering theconstraints calculated in the other above equations.Our search was based on the methods described

in [Sprott, 2010] (this method was first used in[Sprott, 1994], and have been used in many otherresearches such as [Jafari et al., 2013; Molaie et al.,2013; Jafari & Sprott, 2013; Barati et al., 2016;Sprott et al., 2015; Jafari et al., 2016b; Jafari et al.,2016c]). We used our own custom software. Ourobjective was to find the algebraically simplest caseswhich cannot be further reduced by the removal ofterms without destroying the chaos. So we did anexhaustive computer search considering many thou-sands of combinations of the coefficients and ini-tial conditions subject to the constraints, seekingcases for which the largest Lyapunov exponent isgreater than 0.001. For each case that was found,the space of coefficients was searched for values thatare deemed “elegant” in [Sprott, 2010], by which wemean that as many coefficients as possible are setto zero with the others set to ±1 if possible or oth-erwise to a small integer or decimal fraction withthe fewest possible digits.

The simplest cases obtained from the searchprocedure (which may be the simplest possibleNEMs) are listed in Table 1.

3. Dynamical Analysis

In this section, we perform routine dynamical anal-ysis of the designed maps. Strange attractors of thecases in Table 1 are plotted in Fig. 2. Bifurcation,Lyapunov exponent and Kaplan–Yorke diagramsshow different behaviors of the system when chang-ing its parameters. For NEM1, Fig. 3(a) shows thebifurcation diagram with the change of the param-eter a, Fig. 3(b) is Kaplan–Yorke diagram, andFig. 3(c) represents the Lyapunov exponents dia-gram which is derived using the Wolf’s algorithm[Wolf et al., 1985]. The same plots for NEM2 canbe seen in Fig. 4. It can be seen there exists com-mon period-doubling route to chaos. Figure 5 rep-resents the basin of attraction of NEM1. It is clearthat both cases in Table 1 belong to the systemswith hidden attractors, because there is no equi-librium which can have intersection with the basin

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(a) (b)

Fig. 2. Strange attractors of the cases in Table 1: (a) NEM1 and (b) NEM2. Note that the two separated sets of points belongto one attractor consisting of two parts.

(a)

(b)

(c)

Fig. 3. (a) Lyapunov exponents diagram, (b) Kaplan–Yorke diagram and (c) bifurcation diagram of the NEM1 with respectto the parameter b.

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(a)

(b)

(c)

Fig. 4. (a) Lyapunov exponents diagram, (b) Kaplan–Yorke diagram and (c) bifurcation diagram of the NEM2 with respectto the parameter c.

Fig. 5. Basin of attraction of the NEM1. Initial conditions in the light blue region lead to the chaotic attractor, and unboundedregions are shown in yellow.

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of strange attractor. As shown, initial conditions inthe light blue region lead to the chaotic attractor,and unbounded regions are shown in yellow.

4. Conclusion

In this paper, we have tried to design the simplesttwo-dimensional chaotic maps with no equilibria.Such maps belong to the hidden attractors’ categorywhich is a newly introduced category of dynami-cal systems. First, we gave a detailed explanationabout the procedure of designing such systems withquadratic nonlinearities and continuous right-handside functions. Then some dynamical analysis wasinvestigated by the help of plotting return map,bifurcation diagram, Lyapunov exponents diagram,and basin of attraction.

Acknowledgments

Sajad Jafari and Shirin Panahi were supportedby Iran National Science Foundation (Grant No.96000815). This paper is dedicated to the memoryof Professor Gennady A. Leonov.

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1 Introduction2 New No Equilibrium Maps3 Dynamical Analysis4 Conclusion